90 men are engaged to do a piece of work in 40 days but it is found that in 25 days, 2/3 work is completed. How many men should be allowed to go off, so that the work may be finished in time?

Options

A. 10
B. 15
C. 20
D. 25

Answer

To solve the problem step by step, we will use the concept of work done, which is based on the number of men, the number of days, and the total work.

### Step 1: Understand the total work and the work done in the given time.
- **Total work (W)** can be calculated based on the initial information. If 90 men can complete the work in 40 days, the total work can be expressed as:
\[ W = \text{Men} \times \text{Days} = 90 \times 40 = 3600 \text{ man-days} \]

### Step 2: Calculate the work done in 25 days.
- In 25 days, it is given that \( \frac{2}{3} \) of the work is completed. Therefore, the work done in 25 days is:
\[ W_1 = \frac{2}{3} \times 3600 = 2400 \text{ man-days} \]

### Step 3: Determine the remaining work.
- The remaining work \( W_2 \) can be calculated as:
\[ W_2 = W - W_1 = 3600 - 2400 = 1200 \text{ man-days} \]

### Step 4: Calculate the remaining days to complete the work.
- Since 25 days have already been used, the remaining days \( D_2 \) to finish the work is:
\[ D_2 = 40 - 25 = 15 \text{ days} \]

### Step 5: Set up the equation for the remaining work.
- Let \( x \) be the number of men who will go off. The number of men remaining will be \( 90 - x \). The equation for the remaining work can be set up as:
\[ (90 - x) \times 15 = 1200 \]

### Step 6: Solve for \( x \).
1. Expand the equation:
\[ 1350 - 15x = 1200 \]
2. Rearranging gives:
\[ 90 - 15x = 1200 \]
3. Now, isolate \( x \):
\[ 15x = 90 - 1200 \]
\[ 15x = -1110 \]
\[ x = \frac{1110}{15} = 74 \]

### Step 7: Conclusion.
- Therefore, the number of men that should be allowed to go off is:
\[ x = 74 \]